Let us now prove two useful properties of \(\alpha \) -Rank-collections.
First, a vNM function for a given preference relation \(\succeq ^{\text{vNM}}\)  is not unique, however, it is up to positive affine transformations. More specifically, two vNM functions \(U, \tilde{U}\)  represent the same preference relation if and only if there exist \(a >0\)  and \(b \in \mathbb {R}\)  such that \(\tilde{U} = aU + b\) . Due to this property, [1]} argue that any game-theoretic solution concept should be invariant to positive affine transformations of the players' utilities. The following theorem establishes this result for \(\alpha \) -Rank-collections if the utility functions are linear in the types.
